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I shall do my best to make this writeup educational, interesting and fun. And it will include some probability in it, but have no phear, I'll try to make it simple so just about everyone can understand.

Introduction to casinos, probability and life

What are casinos all about? Basically, like accountants, ebay and the porn industry, they want to make money. And (let's assume that the casino is an honest one) - how do they do it? Simple, by making the odds slightly in their favor. And they rely heavily on a wonderful thing called statistics, or probability. In every game in the casino, the odds are tipped in their favor. But it is not overly so, so you almost have an equal chance of winning or losing. Almost. But the more you gamble, the larger are the odds you'll eventually lose. I'm not going to explain the odds on all casino games, but in most it is pretty obvious that odds are against you. In blackjack*, for example, the odds depend on the system used. With one deck only and the option of early surrender, the odds are in the favor of the player, if he is good. So casinos never have blackjack with only one deck, and as far as I am aware, early surrender is no longer an option in casinos. When the dealer hits on a soft 17, the odds improve in favor of the casino. (Rules like the fact that if both you and the dealer bust, you still lose help tilt the odds in the casino's favor). Atlantic City casinos have different rules from Reno casinos, for example, but the odds are always in favor of the house. The dealer will always hit on a 16 and stand on a hard 17 because, probability wise, that gives the best chances of winning. There are many additional rules, and it's a bit complicated, but the house wins on average. Blackjack is a game of skill, and the better you are at it, the less you are likely to lose. In fact, with card counting, it is possible in some circumstances to tilt the odds in your favor (especially with taking insurance), depending on the rules and number of decks. That is why casinos never have blackjack played with only one deck. Overall, the odds are always in the casino's favor. The casino makes sure of that.

The Roulette and expected value

My favorite casino game, and the one which I'll talk about here is the roulette. It is also the easiest game to calculate the odds for. There are 36 numbers (1-36) which are red or black, and one number (0) which is green (in North America, there's also 00, but I won't get into that). And in this little green number lies the secret of the casino's roulette profits. All the odds on the table are calculated as if there are 36 numbers, while there are actually 37. For example, if you place a bet on a single number, and it comes up, you get 36 times your initial bet. Which means the expected value of a bet is 1*(36/37) - 35*(1/37) = -1/37.

No phear, I shall explain forthwith the meaning of expected value, in layman's terms. It is basically the average. It is calculated by multiplying, for every event, its probability by the value, and adding them up. So for example, if you throw a die, what is the expected value? The probabilty of each face is equal, i.e. 1/6, and the values are 1,2,3,4,5,6, so the expected value is 1*(1/6) + 2*(1/6) + 3*(1/6) + 4*(1/6) + 5*(1/6) + 6*(1/6) = 3.5. So the expected value of throwing a die is 3.5 (of course you can't throw 3.5 on a die, but on average, that's what you'd get). And the cool thing about it is that by the laws of large numbers - the more 'samples' there are, the more the average approaches the expected value. So if you throw a die once, you might get any number (1-6), so your average could be 1 or 6. But if you throw 1000000 dice, you'll get an average that's pretty darn close to 3.5, with an absurdly high probability. And that's why the casinos win. You have an expected value of -1/37 for every dollar you bet. Say 37 million dollars are gambled on the roulette table - some will win, some will lose, but the casino will win an amount very close to 1 million dollars.

If you bet on a color, you get even odds (I like that phrase) - you get back double your bet. There are 18 numbers with your color, 18 with the other color, and one green (also not your color). If you bet one dollar, your expected value is 1*(18/37) - 1* (19/37) = (surprise surprise) -1/37. So for every dollar you gamble, you will lose on average 1/37. No matter how you bet. Pretty cool. For the casino, anyway.

How to win every time

So now you know all about probability and expected values and such - let me teach you a trick - how you can win every time. Let's use probability to our advantage here. What are the chances of you not winning? 19/37. Let's get even worse odds, and show it works there. If it works with worse odds, it will work with better odds too. Let's assume you have a 1/10th chance of winning. Very small indeed. But if you keep betting indefinitely, you WILL win, at least once. And we can easily prove it. What are the chances of you not winning the first bet? 9/10. On two bets? 9/10 * 9/10 = 81/100. On n bets? 9n/10n. The mathematically inclined will see that 9n/10n ->0 as n approaches infinity. But even if you have no idea what that last sentence means, you can easily see that this gets smaller for every n (81/100 is less than 9/10, and it gets smaller and smaller). And with many bets, the chances of you not winning any of them is virtually zero. With an infinite amount of bets, the probability is zero (A note to mathematicians - please, let's not get into discussions of infinity. We understand each other. Thank you). So you have a probability of 1 to win sometime. (Probabilities go between 0 and 1. 0 means 'never', and 1 means 'always'). You can try it easily with a die - what are the chances you'll never throw 1 again on a given die? 0. Great, but what good is this particular piece of information? Aha! With each bet, double your bet !

Start by betting a dollar. If you win, stop - you won a dollar. If you lost - bet 2. If you win, you will have 4 dollars, and you had bet 3 in total. If you lose, bet 4 dollars. If you win you will have 8, and you had bet 1 + 2 + 4 = 7 overall. Keep doubling the bet until you win. When you do, you will have won a dollar. Of course, you can start by betting 100, and then you will have won 100 dollars.

Why the casino has an upper limit for bets

The problem is - the casino knows this trick too. So they have a maximum bet. In this world, it doesn't make much of a difference, I suppose, as there is a finite amount of money, so there is a finite amount of bets in any case. But in any casino, even in a world with infinite money, there would be a maximum bet, only so that you wouldn't be able to win.

But the limit could be so high, you may say. You're bound to win before you reach it. But what if you don't? The chances of you losing are small, but if you do, you've lost a LOT of money. And this sets the expected value of your winnings nice and right. Let's see, shall we. Say you bet 1 dollar, then 2, then 4, then 8, then 16, then 32 until you win. If you win, you have won 1 dollar. If you lose, you have lost 63. The chances of you losing are 196/376 = 0.018. So your chances of winning are 1-0.018 = 0.982. Pretty good!!! But your expected value is 1*0.982 - 63*0.018 = 0.982 - 1.15 = -0.17. It doesn't matter what the house limit is or what betting strategy you have, your expected value will always be negative.

Conclusion

I think you will all agree that there is only one conclusion possible - drink up those free drinks.



*I had previously written that a tie wins for the house. I'm sorry - I played Blackjack in a casino once, about 10 years ago, and I had forgotten some of the rules. A tie is a tie, and nobody wins. After some research into blackjack, I rewrote that paragraph, so it should be pretty accurate now.

Also, if you're intrested in reading more about this side of statistics, you may want to look into Martingale System and St. Petersburg paradox

kalen says: The "system" for doing what you suggest is called "Double on Losses" and last I checked you could pay any amount of money for a "pro" to sell you this "secret". Yes- it's still being used. Sigh.

Sources:
www.casinocity.com
www.blackjackrulesonline.com

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